<!DOCTYPE html>
<html lang="en">
  <head>
  <meta charset="utf-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="author" content="Zhou Wei <zromyk@163.com>">
  <title>数学考研-高等数学</title>
  <link rel="shortcut icon" href="/favicon.ico">
  <link rel="stylesheet" href="/style/pure.css">
  <link rel="stylesheet" href="/style/main.css">
  <link rel="stylesheet" href="https://cdn.staticfile.org/font-awesome/4.7.0/css/font-awesome.css">
  <link href="https://apps.bdimg.com/libs/highlight.js/9.1.0/styles/default.min.css" rel="stylesheet">
  <script src='/style/baidu.js'></script>
</head>
<body>
  <div id="menu-background"></div>
  <div id="menu">
    <div class="pure-menu pure-menu-horizontal">
  <div id="menu-block">
    <ul class="pure-menu-list">
      <a class="pure-menu-heading" href="/index.html">ZROMYK</a>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/index.html">主页</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/public/archive/index.html">归档</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/public/download/index.html">下载</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/public/feedback/index.html">反馈</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="/public/about/index.html">关于我</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" href="https://github.com/zromyk"><i class="fa fa-github" style="font-size:32px"></i></a>
</li>

    </ul>
  </div>
</div>

  </div>
  <div id="layout">
    <div class="content">
      <div id="content-articles">
  <h1 id="数学考研-高等数学" class="content-subhead">数学考研-高等数学</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/数学考研-高等数学.html" class="leancloud_visitors" style="display:none" data-flag-title="数学考研-高等数学"></span>
  </p>
  <h1 id="_1">第一部分 高等数学</h1>
<h2 id="1">第1讲 高数预备知识</h2>
<h4 id="1-a_1-dd-neq-0">1. 等差数列（首项  <script type="math/tex"> a_1 </script> ，公差  <script type="math/tex"> d(d \neq 0) </script>  ）</h4>
<p>通项公式</p>
<p>
<script type="math/tex; mode=display">
a_n = a_1 + (n-1)d
</script>
</p>
<p>前  <script type="math/tex"> n </script>  项的和</p>
<p>
<script type="math/tex; mode=display">
S_n = \cfrac{n}{2}(a_1+a_n)
</script>
</p>
<h4 id="2-a_1-rr-neq-0">2. 等比数列（首项  <script type="math/tex"> a_1 </script> ，公比  <script type="math/tex"> r(r \neq 0) </script> ）</h4>
<p>通项公式</p>
<p>
<script type="math/tex; mode=display">
a_n = a_1r^{(n-1)}
</script>
</p>
<p>前  <script type="math/tex"> n </script>  项的和</p>
<p>
<script type="math/tex; mode=display">
S_n = 
\begin{cases}
na_1, & \text{r = 1} \\[2ex]
\cfrac{a_1(1-r^n)}{1-r}, & r \neq 1
\end{cases}
</script>
</p>
<p>常用  <script type="math/tex"> 1 + r + r^2 + \cdots + r^{n-1} = \cfrac{1 - r^n}{1 - r} </script>
</p>
<h4 id="3">3. 三角函数基本关系</h4>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学考研-高等数学.assets/三角函数.jpg" alt="sin_cos" style="zoom:33%;" /></p>
<h5 id="1_1">1）倍角公式</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin2\alpha &= 2\sin\alpha *\cos\alpha \\[2ex]
\cos2\alpha &=\cos^2\alpha -\sin^2\alpha \\ 
\quad &= 1 - 2\sin^2\alpha \\ 
\quad &= 2\cos^2\alpha - 1 \\[2ex]
\tan2\alpha &= \cfrac{2 \tan\alpha}{1 - \tan^2\alpha}
\end{split}\end{equation}
</script>
</p>
<h5 id="2">2）和差公式</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin(\alpha \pm \beta) &=\sin\alpha\cos\beta \pm\cos\alpha\sin\beta \\
\sin(\alpha \pm \beta) &=\cos\alpha\cos\beta \mp\sin\alpha\sin\beta \\
\tan(\alpha \pm \beta) &= \cfrac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}
\end{split}\end{equation}
</script>
</p>
<h5 id="3_1">3）积化和差公式</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin\alpha\cos\beta &= \frac{1}{2}[\sin(\alpha + \beta) +\sin(\alpha - \beta)] \\
\cos\alpha\sin\beta &= \frac{1}{2}[\sin(\alpha + \beta) -\sin(\alpha - \beta)] \\
\sin\alpha\sin\beta &= \frac{1}{2}[\cos(\alpha + \beta) +\cos(\alpha - \beta)] \\
\cos\alpha\cos\beta &= \frac{1}{2}[\cos(\alpha - \beta) -\cos(\alpha + \beta)]
\end{split}\end{equation}
</script>
</p>
<h5 id="4">4）和差化积公式</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin\alpha +\sin\beta &= \quad 2\sin\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2} \\
\sin\alpha -\sin\beta &= \quad 2\sin\frac{\alpha - \beta}{2}\cos\frac{\alpha + \beta}{2} \\
\cos\alpha +\cos\beta &= \quad 2\cos\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2} \\
\cos\alpha -\cos\beta &= -\ 2\sin\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}
\end{split}\end{equation}
</script>
</p>
<h4 id="4_1">4. 因式子分解公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
(a + b)^n &= C_n^0a^n + C_n^1a^{n-1}b + ... + C_n^nb^n \\[2ex]
a^n - b^n &= (a - b)(a^n + a^{n-1}b + \cdots + ab^{n-1} + b^n) \\
\end{split}\end{equation}
</script>
</p>
<h4 id="5">5. 常用不等式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\vert\vert a \vert - \vert b \vert\vert &\le \vert a \pm b \vert \le \vert a \vert + \vert b \vert \\[2ex]
\cfrac{2}{\frac{1}{a}+\frac{1}{b}} \le \sqrt{ab} &\le \frac{a+b}{2} \le \sqrt{\frac{a^2+b^2}{2}} \ (a,b \gt 0) \\[2ex]
\cfrac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}} \le\sqrt[3]{abc} &\le \frac{a+b+c}{3} \le \sqrt{\frac{a^2+b^2+c^2}{3}} \ (a,b,c \gt 0) \\[2ex]
\sin\ x &\lt x \lt tan\ x \ (0 \lt x \lt \frac{\pi}{2})
\end{split}\end{equation}
</script>
</p>
<h2 id="2_1">第2讲 函数极限</h2>
<h4 id="0">0. 基本极限公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\lim_{x\to\infty}(1+\cfrac{1}{x})^x &= e ⟺ \lim_{x\to0}(1+x)^{\frac{1}{x}} = e \\
\lim_{x\to\infty}(1+\cfrac{a}{x})^{bx} &= e^{ab}
\end{split}\end{equation}
</script>
</p>
<h4 id="1_2">1. 洛必达法则</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\lim_{x\to·}\cfrac{f(x)}{g(x)} &= \lim_{x\to·}\cfrac{f'(x)}{g'(x)} \\[1em]
\lim_{x\to·}\cfrac{\int_{a}^{x}f(t)dt}{\int_{a}^{x}g(t)dt} &= \lim_{x\to·}\cfrac{f(x)}{g(x)} \\[1em]
\lim_{x\to·}\cfrac{\int_{\psi(x)}^{\phi(x)}f(t)dt}{\int_{\psi(x)}^{\phi(x)}g(t)dt} &= \lim_{x\to·}\cfrac{f[\phi(x)]\phi'(x)-f[\psi(x)]\psi'(x)}{g[\phi(x)]\phi'(x)-g[\psi(x)]\psi'(x)}
\end{split}\end{equation}
</script>
</p>
<h4 id="2_2">2. 泰勒公式</h4>
<video style="border: 1px solid rgba(0, 0, 0, 1);" controls="controls" width="100%" src="/post/数学考研-高等数学.assets/泰勒级数.mov"></video>

<h5 id="5_1">定理5（泰勒公式）（忘记常用泰勒级数展开式，可以用下面的公式求）</h5>
<p>
<script type="math/tex; mode=display">
f(x) = f(x_0) + f'(x_0)(x-x_0) + ... + \cfrac{1}{n!}f^{(n)}(x_0)(x-x_0)^n + \cfrac{1}{(n+1)!}f^{(n)}(\xi)(x-x_0)^{n+1}+...
</script>
</p>
<p>其中 <script type="math/tex"> \xi </script> 介于 <script type="math/tex"> x,x_0 </script> 之间.<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin x &= \sin x_0 + (\cos x_0)(x - x_0) + \cfrac{1}{2!}(-\sin x_0)(x - x_0)^2 + \cfrac{1}{3!}(-\cos x_0)(x - x_0)^3 + ... \\
       &= x - \cfrac{x^3}{3!} + ...
\end{split}\end{equation}
</script>
</p>
<h4 id="_2">常用泰勒级数展开式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
   \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + ... = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n}}{(2n)!} \\[1ex]
  \cosh x &= 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + ... = \sum_{n=0}^\infty\cfrac{x^{2n}}{(2n)!} \\[1ex]
\arccos x &= \cfrac{\pi}{2} - \arcsin x \\[2em]
   \sin x &= x - \frac{x^3}{3!} + ... = \sum_{n=0}^\infty(-1)^n\cfrac{x^{2n+1}}{(2n+1)!} \\[1ex]
  \sinh x &= x + \frac{x^3}{3!} + ... = \sum_{n=0}^\infty\cfrac{x^{2n+1}}{(2n+1)!} \\[1ex]
\arcsin x &= x + \frac{x^3}{6} + o(x^3) \\[2em]
   \tan x &= x + \frac{x^3}{3} + o(x^3) \\[1ex]
\arctan x &= x - \frac{x^3}{3} + o(x^3) \\[2em]
 \ln(1+x) &= x - \frac{x^2}{2} + \frac{x^3}{3} ... = \sum_{n=1}^\infty(-1)^{n-1}\cfrac{x^{n}}{n},-1\lt x\le 1 \\[1ex]
      e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... = \sum_{n=0}^\infty\cfrac{x^{n}}{n!} \\[2ex]
  (1+x)^a &= 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3 + o(x^3) \\[2em]
\cfrac{1}{1-x} &= \sum_{n=0}^\infty x^n,|x|\lt1 \\[1ex]
\cfrac{1}{1+x} &= \sum_{n=0}^\infty (-1)^nx^n,|x|\lt1
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<ol>
<li>
<p>无穷级数求和函数，应灵活运用上面的泰勒级数展开式.</p>
</li>
<li>
<p>在与导数相关题目中的应用：</p>
</li>
</ol>
<p>【2016年考研数一16题】设函数 <script type="math/tex">f(x)=\arctan x-\cfrac{x}{1+ax^2}</script>，且 <script type="math/tex">f'''(0)=1</script>，则 <script type="math/tex">a=</script> ____<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
f(x)&=\arctan x-\cfrac{x}{1+ax^2} \\
&=x-\cfrac{1}{3}x^3+o(x^3)-x(1-ax^2+o(x^2)) \\
&=(a-\cfrac{1}{3})x^3+o(x^3) \\[1ex]
f'''(x)&=3*2*1*(a-\cfrac{1}{3})=1 \\[1ex]
a&=\cfrac{1}{2}
\end{split}\end{equation}
</script>
</p>
</blockquote>
<h4 id="3_2">3. 无穷比介小</h4>
<ol>
<li>若  <script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = 0 </script>  则称  <script type="math/tex"> \alpha(x) </script>  是  <script type="math/tex"> \beta(x) </script>  高阶无穷小（高阶趋向于0的速度更快），记  <script type="math/tex"> \alpha(x) = o(\beta(x)) </script>
</li>
<li>若  <script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = \infty </script>  则称  <script type="math/tex"> \alpha(x) </script>  是  <script type="math/tex"> \beta(x) </script>  低阶无穷小</li>
<li>若  <script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = c \neq 0 </script>  则称  <script type="math/tex"> \alpha(x) </script>  是  <script type="math/tex"> \beta(x) </script>  同阶无穷小（趋向于0的速度相近）</li>
<li>若  <script type="math/tex"> \lim\cfrac{\alpha(x)}{\beta(x)} = 1 </script>  则称  <script type="math/tex"> \alpha(x) </script>  是  <script type="math/tex"> \beta(x) </script>  等阶无穷小（趋向于0的速度相等），记  <script type="math/tex"> \alpha(x) ～ \beta(x) </script>
</li>
<li>若  <script type="math/tex"> \lim\cfrac{\alpha(x)}{[\beta(x)]^k} = c \neq 0，k \gt 0 </script>  则称  <script type="math/tex"> \alpha(x) </script>  是  <script type="math/tex"> \beta(x) </script>
<script type="math/tex"> k </script>  阶无穷小</li>
</ol>
<h4 id="4_2">4.  无穷小的运算规则</h4>
<ol>
<li>有限个无穷小的和是无穷小</li>
<li>有限个无穷小的乘积是无穷小</li>
<li>有界函数与无穷小的乘积是无穷小</li>
<li>无穷小的运算</li>
<li>加减法：低价吸收高阶  <script type="math/tex"> o(x^2) \pm o(x^3) = o(x^2) </script>
</li>
<li>乘法：阶数累加  <script type="math/tex"> o(x^2) * o(x^3) = o(x^5) </script>
</li>
<li>非0常数相乘不影响阶数   <script type="math/tex"> o(k * x^2) = k * o(x^2) </script>
</li>
</ol>
<h4 id="5_2">5. 常用的等价无穷小</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
   \sin x &\sim x \\[2ex]
   \tan x &\sim x \\[2ex]
\arcsin x &\sim x \\[2ex]
\arctan x &\sim x \\[2ex]
\ln(1 + x) &\sim x \\[2ex]
  e^x - 1 &\sim x \\[2ex]
  a^x - 1 &\sim x\ln a \\[3ex]
1 -\cos\ x &\sim \frac{1}{2}x^2 \\[2ex]
(1 + x)^a - 1&\sim ax
\end{split}\end{equation}
</script>
</p>
<h4 id="6">6. 函数的连续与间断</h4>
<ol>
<li>可去间断点</li>
<li>跳跃间断点</li>
<li>无穷间断点</li>
<li>震荡间断点</li>
</ol>
<h3 id="_3">补充：渐近线</h3>
<h4 id="1_3">1、水平渐近线和铅直渐近线</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{x\to\infty}[f(x)] &= y_0,\ 则y=y_0为水平渐近线 \\[2ex]
实际上是求 \lim_{x\to\infty}[f(x)-y_0] &= 0 \\[1em]
\lim_{x\to x_0}[f(x)]&=\infty,\ 则x=x_0为铅直渐近线 
\end{split}\end{equation}
</script>
</p>
<h4 id="2x">2、斜渐近线的正确求法(在x趋向于无穷时)</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{x\to\infty}(\cfrac{f(x)}{x}) &= A \\[1ex]
\lim_{x\to\infty}[f(x)-Ax] &= B \\[1ex]
渐近线方程为\ y &= Ax + B \\[1em]
实际上是求\ \lim_{x\to\infty}[f(x)-(Ax+B)] &= 0
\end{split}\end{equation}
</script>
</p>
<h2 id="4_3">第4讲 一元函数微分学</h2>
<h5 id="1_4">1. 导数定义</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f'(x_0) &= \lim_{\Delta x \to 0}\frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} \\[3ex]
&= \lim_{x \to x_0}\frac{f(x) - f(x_0)}{x - x_0} \\
\end{split}\end{equation}
</script>
</p>
<p>无穷导数  <script type="math/tex"> \infty </script>  视为倒数不存在</p>
<blockquote class="content-quote">
<p>常用性质：<script type="math/tex"> f(x) </script> 为偶函数 <script type="math/tex"> \Rightarrow f'(x) </script> 为奇函数 <script type="math/tex"> \Rightarrow f''(x) </script> 为偶函数 <script type="math/tex"> \Rightarrow f^{(3)}(x) </script> 为奇函数&hellip;</p>
</blockquote>
<h5 id="2_3">2. 导数与微分的计算</h5>
<p>积的导数： <script type="math/tex"> [u(x)v(x)]' = u'(x)v(x) + u(x)v'(x) </script>
</p>
<p>积的微分： <script type="math/tex"> d[u(x)v(x)] = du(x)v(x) + u(x)dv(x) </script>
</p>
<p>商的导数： <script type="math/tex"> [\cfrac{u(x)}{v(x)}]' = \cfrac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}, v(x) \neq 0 </script>
</p>
<p>商的微分： <script type="math/tex"> d[\cfrac{u(x)}{v(x)}] = \cfrac{du(x)v(x) - u(x)dv(x)}{[v(x)]^2}, v(x) \neq 0 </script>
</p>
<p>复合函数的导数： <script type="math/tex"> \{f[g(x)]\}' = f'[g(x)]g'(x) </script>
</p>
<p>复合函数的微分： <script type="math/tex"> d\{f[g(x)]\} = f'[g(x)]g'(x)dx </script>
</p>
<p><strong>注意：</strong> <script type="math/tex"> \{f[g(x)]\}' = \cfrac{d\{f[g(x)]\}}{dx}, f'[g(x)] = \cfrac{d\{f[g(x)]\}}{dg(x)} </script>
</p>
<p><strong>微分的定义：</strong> <script type="math/tex"> dy = f'(x_0) \Delta x </script>
</p>
<h5 id="3_3">3. 可微、可导、连续、可积的关系</h5>
<p>可微 ⟺ 可导 ⟹ 连续 ⟹ 可积</p>
<p>可导：（左极限要等于右极限） <script type="math/tex"> y = |x|, \lim\limits_{x \to 0^-} = -1, \lim\limits_{x \to 0^+} = 1, 在\ x = 0\ 处不可导 </script>
</p>
<h5 id="4_4">4. 莱布尼茨公式</h5>
<p>
<script type="math/tex; mode=display">
(uv)^{(n)} = u^{(n)}v + C^1_nu^{(n-1)}v' + \cdots + C^{n-1}_nu'v^{(n-1)} + uv^{(n)}
</script>
</p>
<h5 id="5_3">5. 泰勒公式</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n \\[2ex]
f(x) &= \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n
\end{split}\end{equation}
</script>
</p>
<h5 id="6_1">6. 参数方程</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{dy}{dx} &= \cfrac{\cfrac{dy}{dt}}{\cfrac{dx}{dt}} = \cfrac{\psi'(t)}{\varphi'(t)} \\[2ex]
\cfrac{d^2y}{dx^2} &= \cfrac{d(\cfrac{dy}{dx})}{dx} = \cfrac{\cfrac{d(\cfrac{dy}{dx})}{dt}}{\cfrac{dx}{dt}}= \cfrac{\psi''(t)\varphi'(t)-\psi'(t)\varphi''(t)}{[\varphi'(t)]^3}
\end{split}\end{equation}
</script>
</p>
<ol>
<li>若  <script type="math/tex"> f(x) </script>  是可导的偶函数，则  <script type="math/tex"> f'(x) </script>  是奇函数</li>
<li>若  <script type="math/tex"> f(x) </script>  是可导的奇函数，则  <script type="math/tex"> f'(x) </script>  是偶函数</li>
</ol>
<h2 id="6_2">第6讲 一元函数微分学的应用</h2>
<h5 id="1_5">定理1（费马定理）</h5>
<p>
<script type="math/tex; mode=display">
设f(x)在x_0处满足
\begin{cases}
可导 \\
取极值
\end{cases}
，则f'(x)=0
</script>
</p>
<h5 id="2_4">定理2（罗尔定理）</h5>
<p>
<script type="math/tex; mode=display">
设f(x)在x_0处满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导 \\
f(a) = f(b)
\end{cases}
，则存在\xi\in(a,b)，使得f'(\xi)=0
</script>
</p>
<h5 id="3_4">定理3（拉格朗日中值定理）</h5>
<p>
<script type="math/tex; mode=display">
设f(x)在满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导
\end{cases}
，则存在\xi\in(a,b)，使得 \\
f(b) - f(a) = f'(\xi)(b - a) \\ 
即f'(\xi) = \cfrac{f(b) - f(a)}{b - a}
</script>
</p>
<h5 id="4_5">定理4（柯西中值定理）</h5>
<p>
<script type="math/tex; mode=display">
设f(x),g(x)在满足
\begin{cases}
[a,b]上连续 \\
(a,b)内可导 \\
g'(x)\ne0
\end{cases}
，则存在\xi\in(a,b)，使得 \\ 
\cfrac{f'(\xi)}{g'(\xi)} = \cfrac{f(b) - f(a)}{g(b) - g(a)}
</script>
</p>
<h5 id="5_4">定理5（泰勒公式）</h5>
<p>
<script type="math/tex; mode=display">
f(x) = f(x_0) + f'(x_0)(x-x_0) + ... + \cfrac{1}{n!}f^{(n)}(x_0)(x-x_0)^n + \cfrac{1}{(n+1)!}f^{(n)}(\xi)(x-x_0)^{n+1}
</script>
</p>
<p>其中 <script type="math/tex"> \xi </script> 介于 <script type="math/tex"> x,x_0 </script> 之间.<br />
<script type="math/tex; mode=display">
\sin x = \sin0 + (\cos0)(x - 0)
</script>
</p>
<h5 id="6_3">定理6（积分中值定理）</h5>
<p>
<script type="math/tex; mode=display">
设f(x)在[a,b]上连续，则存在\xi\in[a,b]，使得 \\ 
\int_{a}^{b}f(x)dx = f(\xi)(b - a)
</script>
</p>
<h2 id="9">第9讲 一元函数积分学</h2>
<h5 id="_4">“点火公式”</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_0^{\frac{\pi}{2}}\sin^nx\ dx &= \int_0^{\frac{\pi}{2}}\cos^nx\ dx = 
\begin{cases}
\cfrac{n-1}{n}·\cfrac{n-3}{n-2}\cdots\cfrac{2}{3}，n为大于1的奇数\\[2ex]
\cfrac{n-1}{n}·\cfrac{n-3}{n-2}\cdots\cfrac{1}{2}·\cfrac{\pi}{2}，n为偶数
\end{cases}
\end{split}\end{equation}
</script>
</p>
<h5 id="_5">伽马函数</h5>
<p>
<script type="math/tex; mode=display">
实数域：\Gamma(x) = \int_0^{+\infty}t^{x-1}e^{-t}dt,\ (x>0)
</script>
</p>
<p>伽马函数的推导<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{1}{1-x} &= \sum_{n=0}^\infty x^n (对比)\\[1ex]
&= \int_0^{+\infty}e^{-(1-x)t}dt \\[1ex]
&= \int_0^{+\infty}e^{-t+xt}dt \\[1ex]
&= \int_0^{+\infty}e^{-t}\sum_{n=0}^\infty \cfrac{(xt)^n}{n!}dt \\[1ex]
&= \sum_{n=0}^\infty \cfrac{\int_0^{+\infty}t^ne^{-t} dt}{n!}x^n (对比)\\[2em]
\int_0^{+\infty}t^ne^{-t} dt &= n!\ \ \ \ (便利公式)
\end{split}\end{equation}
</script>
</p>
<h5 id="_6">分部积分</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int u(x)v'(x)dx &= \int u(x)dv(x) \\
&= u(x)v(x) - \int u'(x)v(x)dx
\end{split}\end{equation}
</script>
</p>
<h5 id="_7">基本公式</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
            \int\cfrac{1}{x^2}dx &= -\cfrac{1}{x} + C \\[1ex]
       \int\cfrac{1}{\sqrt{x}}dx &= 2\sqrt{x} + C \\[2em]
                      \int e^xdx &= e^x + C \\[1ex]
                      \int a^xdx &= \cfrac{a^x}{\ln a} + C \\[2em]
          \int\cfrac{1}{1+x^2}dx &= \arctan x + C \\[1ex]
        \int\cfrac{1}{a^2+x^2}dx &= \cfrac{1}{a}\arctan\cfrac{x}{a} + C(a\gt0)\\[2em]
   \int\cfrac{1}{\sqrt{1+x^2}}dx &= \arcsin x + C \\[1ex]
 \int\cfrac{1}{\sqrt{a^2+x^2}}dx &= \arcsin\cfrac{x}{a} + C(a\gt0)\\[2em]
 \int\cfrac{1}{\sqrt{x^2+a^2}}dx &= \ln(x+\sqrt{x^2+a^2}) + C \\[1ex]
 \int\cfrac{1}{\sqrt{x^2-a^2}}dx &= \ln|x+\sqrt{x^2-a^2}|  + C(|x|\gt|a|)\\[2em]
                   \int\sin^2xdx &= \cfrac{x}{2} - \cfrac{\sin 2x}{4} + C \\[1ex]
                   \int\cos^2xdx &= \cfrac{x}{2} + \cfrac{\sin 2x}{4} + C \\[1ex]
                   \int\tan^2xdx &= \tan x - x + C
\end{split}\end{equation}
</script>
</p>
<h2 id="13">第13讲 多元函数微分学</h2>
<h3 id="1_6">1. 导数与微分</h3>
<p>
<script type="math/tex; mode=display">
偏导数存在 \Rightarrow 可微
\begin{cases}
\Rightarrow 偏导数存在（某方向双侧）\\[2ex]
\Rightarrow 连续 \Rightarrow 极限存在（全方向） \\[2ex]
\Rightarrow 方向导数存在（某方向单侧）
\end{cases}
</script>
</p>
<h5 id="1_7">1）偏导数的定义公式</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f'_x(x,y)
&= \lim_{\Delta x\to0}\cfrac{f(x_0 + \Delta x, y_0) - f(x_0, y_0)}{\Delta x} \\[3ex]
&= \lim_{x \to x_0}\frac{f(x,y_0) - f(x_0,y_0)}{x - x_0} \\
\end{split}\end{equation}
</script>
</p>
<h5 id="2_5">2）二元函数微分的定义</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f(x,y)=&f(x_0,y_0)+f'_x(x_0,y_0)\Delta x+f'_y(x_0,y_0)\Delta y+o(\sqrt{(\Delta x)^2+(\Delta y)^2})\\[3ex]
=&f(x_0,y_0)+f'_x(x_0,y_0)(x-x_0)+f'_y(x_0,y_0)(y-y_0)+o(\sqrt{(x-x_0)^2+(y-y_0)^2})
\end{split}\end{equation}
</script>
</p>
<p>可微的条件<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
（可微的条件）&\lim_{\Delta x \to 0,\Delta y \to 0}\cfrac{\Delta z - [f'_x(x_0,y_0)\Delta x + f'_y(x_0,y_0)\Delta y]}{\sqrt{(\Delta x)^2+(\Delta y)^2}} \\[3ex]
=&\lim_{(x,y)\to(x_0,y_0)}\cfrac{f(x,y)-f(x_0,y_0)-f'_x(x_0,y_0)(x-x_0)-f'_y(x_0,y_0)(y-y_0)}{\sqrt{(x-x_0)^2+(y-y_0)^2}} \\[2ex]
=&\ 0
\end{split}\end{equation}
</script>
</p>
<h5 id="3_5">3）复合函数求导法</h5>
<p>设 <script type="math/tex"> z=z(u,v),u=u(x,y),v=v(x,y) </script>
<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\begin{pmatrix}
\cfrac{\partial z}{\partial x} &\cfrac{\partial z}{\partial y} \\
\end{pmatrix} &=
\begin{pmatrix}
\cfrac{\partial z}{\partial u} &\cfrac{\partial z}{\partial v} \\
\end{pmatrix}
\begin{pmatrix}
\cfrac{\partial u}{\partial x} &\cfrac{\partial u}{\partial y} \\
\cfrac{\partial v}{\partial x} &\cfrac{\partial v}{\partial y} \\
\end{pmatrix} \\[1ex]
\cfrac{\partial z}{\partial x} &= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial x}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial x} \\[1ex]
\cfrac{\partial z}{\partial y} &= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial y}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial y}
\end{split}\end{equation}
</script>
<br />
设 <script type="math/tex"> z=z(u,v),u=u(x),v=v(x) </script> ，即 <script type="math/tex"> z </script> 最终是 <script type="math/tex"> x </script> 的函数，则 <script type="math/tex"> \cfrac{dz}{dx} </script> 叫全导数<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{dz}{dx} &=
\begin{pmatrix}
\cfrac{\partial z}{\partial u} &\cfrac{\partial z}{\partial v} \\
\end{pmatrix}
\begin{pmatrix}
\cfrac{\partial u}{\partial x} \\ 
\cfrac{\partial v}{\partial x}
\end{pmatrix} \\[1ex]
&= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial x}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial x}
\end{split}\end{equation}
</script>
<br />
全微分形式不变性<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
dz &= \cfrac{\partial z}{\partial x}dx + \cfrac{\partial z}{\partial y}dy\\
   &= \cfrac{\partial z}{\partial u}du + \cfrac{\partial z}{\partial v}dv
\end{split}\end{equation}
</script>
</p>
<p>将 <script type="math/tex">z=z(x,y)</script> 转化为 <script type="math/tex">F(x,y,z)=0</script> 求偏导<br />
<script type="math/tex; mode=display">
\cfrac{\partial z}{\partial x} = -\cfrac{F'_x}{F'_z}\\
\cfrac{\partial z}{\partial y} = -\cfrac{F'_y}{F'_z}
</script>
</p>
<h3 id="2_6">2. 二元函数极值</h3>
<p>先求一阶偏导数 <br />
<script type="math/tex; mode=display">
f'_x(x,y) \\
f'_y(x,y)
</script>
</p>
<p>再求二阶偏导数</p>
<p>
<script type="math/tex; mode=display">
A = f''_{xx}(x_0, y_0) \\[1ex]
B = f''_{xy}(x_0, y_0) \\[1ex]
C = f''_{yy}(x_0, y_0)
</script>
</p>
<p>
<script type="math/tex"> (x_0, y_0) </script>  为一阶导数  <script type="math/tex"> f_x(x,y) = 0，f_y(x,y) = 0 </script>  的点</p>
<ol>
<li>
<script type="math/tex"> AC-B^2 \gt 0 </script>  且  <script type="math/tex"> A \gt 0 </script>  极小值点</li>
<li>
<script type="math/tex"> AC-B^2 \gt 0 </script>  且  <script type="math/tex"> A \lt 0 </script>  极大值点</li>
<li>
<script type="math/tex"> AC-B^2 \lt 0 </script>  非极值点（鞍点）</li>
<li>
<script type="math/tex"> AC-B^2 = 0 </script>  不确定</li>
</ol>
<h3 id="3_6">3. 条件极值</h3>
<blockquote class="content-quote">
<p>函数 <script type="math/tex">f(x,y)</script> 在条件 <script type="math/tex">\phi(x,y)=0</script> 下取得极值的必要条件，设 <script type="math/tex">(x_0,y_0)</script> 处为极值点，则<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\phi(x_0,y_0) &= 0 \\[2ex]
df &= f'_x(x,y)dx + f'_y(x,y)dy  \\[2ex]
\Rightarrow &\begin{cases}
\cfrac{df}{dx}_{(x,y)=(x_0,y_0)} = f'_x(x_0,y_0) + f'_y(x_0,y_0)\cfrac{dy}{dx}_{(x,y)=(x_0,y_0)} = 0 \\[2ex]
\cfrac{df}{dy}_{(x,y)=(x_0,y_0)} = f'_y(x_0,y_0) + f'_x(x_0,y_0)\cfrac{dx}{dy}_{(x,y)=(x_0,y_0)} = 0 \\[2ex]
\cfrac{dy}{dx}_{(x,y)=(x_0,y_0)} = -\cfrac{\phi'_x(x_0,y_0)}{\phi'_y(x_0,y_0)} \\[2ex]
\cfrac{dx}{dy}_{(x,y)=(x_0,y_0)} = -\cfrac{\phi'_y(x_0,y_0)}{\phi'_x(x_0,y_0)}
\end{cases} \\[2em]
\Rightarrow &\begin{cases}
\cfrac{df}{dx}_{(x,y)=(x_0,y_0)} =& f'_x(x_0,y_0) -\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}\phi'_x(x_0,y_0) = 0 \\[2ex]
&f'_x(x_0,y_0)-\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)}\phi'_x(x_0,y_0) = 0 \\[2ex]
\cfrac{df}{dy}_{(x,y)=(x_0,y_0)} =& f'_y(x_0,y_0) -\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)}\phi'_y(x_0,y_0) = 0 \\[2ex]
&f'_y(x_0,y_0)-\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}\phi'_y(x_0,y_0) = 0 \\[2ex]
\end{cases} \\[2em]
综上&\begin{cases}
f'_x(x_0,y_0)+\lambda\phi'_x(x_0,y_0) = 0 \\[2ex]
f'_y(x_0,y_0)+\lambda\phi'_y(x_0,y_0) = 0 \\[2ex]
\phi(x_0,y_0) = 0 \\[2ex]
\end{cases}
，且\lambda = -\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)} = -\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}
\end{split}\end{equation}
</script>
<br />
恰好与下式相等</p>
</blockquote>
<h5 id="fxy">（二元）函数 <script type="math/tex">f(x,y)</script> 在条件</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
F(x,y,\lambda)&=f(x,y)+\lambda\phi(x,y)\\[2ex]
\phi(x,y)&=0\\[2em]
\end{split}\end{equation}
</script>
</p>
<p>下取得极值的必要条件：</p>
<p>
<script type="math/tex; mode=display">
\begin{cases}
F'_x(x,y,\lambda) = f'_x(x,y) + \lambda \phi'_x(x,y) = 0 \\[2ex]
F'_y(x,y,\lambda) = f'_y(x,y) + \lambda \phi'_y(x,y) = 0 \\[2ex]
F'_\lambda(x,y,\lambda) = \phi(x,y)= 0 \\
\end{cases}
</script>
</p>
<h5 id="fxyz">（三元）函数 <script type="math/tex">f(x,y,z)</script> 在条件</h5>
<p>
<script type="math/tex; mode=display">
F(x,y,z,\lambda,\mu)=f(x,y,z)+\lambda\phi(x,y,z)+\mu\psi(x,y,z)\\[2ex]
\begin{cases}
\phi(x,y,z)=0 \\[1ex]
\psi(x,y,z)=0
\end{cases}
</script>
</p>
<p>下取得极值的必要条件：</p>
<p>
<script type="math/tex; mode=display">
\begin{cases}
F'_x(x,y,z,\lambda,\mu) = f'_x(x,y,z) + \lambda \phi'_x(x,y,z) + \mu \psi'_x(x,y,z) = 0 \\[1ex]
F'_y(x,y,z,\lambda,\mu) = f'_y(x,y,z) + \lambda \phi'_y(x,y,z) + \mu \psi'_y(x,y,z) = 0 \\[1ex]
F'_z(x,y,z,\lambda,\mu) = f'_z(x,y,z) + \lambda \phi'_z(x,y,z) + \mu \psi'_z(x,y,z) = 0 \\[1ex]
F'_\lambda(x,y,z,\lambda,\mu) = \phi(x,y,z) = 0 \\[1ex]
F'_\mu(x,y,z,\lambda,\mu) = \psi(x,y,z)= 0 \\
\end{cases}
</script>
</p>
<h2 id="15">第15讲 微分方程</h2>
<h4 id="1_8">1. 一阶微分方程的求解</h4>
<h5 id="1_9">1）可分离变量型</h5>
<h5 id="2_7">2）齐次型</h5>
<p>
<script type="math/tex"> y'+p(x)y=0 </script>
</p>
<h5 id="3_7">3）一阶线性型</h5>
<p>能写成 <script type="math/tex"> y'+p(x)y=q(x) </script> ，在方程的两边同时乘以 <script type="math/tex"> e^{\int{p(x)dx}} </script>
<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
e^{\int{p(x)dx}}y'+e^{\int{p(x)dx}}p(x)y &= e^{\int{p(x)dx}}q(x) \\[3ex]
                    [e^{\int{p(x)dx}}y]' &= e^{\int{p(x)dx}}q(x) \\[3ex]
                       e^{\int{p(x)dx}}y &= \int{e^{\int{p(x)dx}}q(x)}+C \\[3ex]
                                       y &= e^{-\int{p(x)dx}}[\int{e^{\int{p(x)dx}}q(x)}+C]
\end{split}\end{equation}
</script>
</p>
<h5 id="_8">伯努利方程</h5>
<p>
<script type="math/tex"> y'+p(x)y=q(x)y^n </script>
</p>
<p>a. 先变形 <script type="math/tex"> \cfrac{y'+p(x)y}{y^n}=\cfrac{q(x)y^n}{y^n}\Rightarrow y^{-n}y'+p(x)y^{1-n}=q(x) </script>
</p>
<p>b. 令 <script type="math/tex"> z=y^{1-n} </script> ，得 <script type="math/tex"> \cfrac{dz}{dx}=(1-n)y^{-n}\cfrac{dy}{dx} </script> ，则 <script type="math/tex"> \cfrac{1}{1-n}\cfrac{dz}{dx}=y^{-n}\cfrac{dy}{dx} </script> ，带入到上面的式子中</p>
<p>c. 得到一阶非齐次微分方程 <script type="math/tex"> \cfrac{1}{1-n}\cfrac{dz}{dx}+p(x)z=q(x) </script> ，即<br />
<script type="math/tex; mode=display">
\cfrac{dz}{dx}+(1-n)p(x)z=(1-n)q(x)
</script>
</p>
<h4 id="2_8">2. 二阶微分方程的求解</h4>
<h5 id="1-ypyqy0">1）齐次线性方程  <script type="math/tex"> y''+py'+qy=0 </script>
</h5>
<p>若 <script type="math/tex"> p^2 - 4q \gt 0 </script> ，设 <script type="math/tex"> \lambda_1,\lambda_2 </script> 是特征方程的两个不等实根，即 <script type="math/tex"> \lambda_1\neq\lambda_2 </script> ，可得其通解为<br />
<script type="math/tex; mode=display">
y = C_1e^{\lambda_1x} + C_2e^{\lambda_2x}
</script>
<br />
若 <script type="math/tex"> p^2 - 4q = 0 </script> ，设 <script type="math/tex"> \lambda_1,\lambda_2 </script> 是特征方程的两个相等实根，即二重根，令 <script type="math/tex"> \lambda_1=\lambda_2=\lambda </script> ，可得其通解为<br />
<script type="math/tex; mode=display">
y = (C_1 + C_2x) e^{\lambda x}
</script>
<br />
若 <script type="math/tex"> p^2 - 4q \lt 0 </script> ，设 <script type="math/tex"> \alpha\pm\beta i </script> 是特征方程的一对共轭复根，可得其通解为<br />
<script type="math/tex; mode=display">
y =e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)
</script>
</p>
<h5 id="2-ypyqyfx">2）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f(x) </script>
</h5>
<p>当 <script type="math/tex"> f(x) = e^{ax}P_n(x) </script> 时，设特解为 <script type="math/tex"> y^{*} = e^{ax}Q_n(x)x^k </script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{cases}
e^{ax}\text{照抄} \\[2ex]
Q_n(x)\text{为x的n次多项式} \\[2ex]
k = 
\begin{cases}
0,\ \alpha\neq\lambda_1,\alpha\ne\lambda_2\\[2ex]
1,\ \alpha=\lambda_1\text{或}\alpha=\lambda_1\\[2ex]
2,\ \alpha=\lambda_1=\lambda_1
\end{cases}
\end{cases}
</script>
</p>
<p>当 <script type="math/tex"> f(x) = e^{ax}[P_m(x)\cos\beta x + P_n(x)\sin\beta x] </script> 时，设特解为 <script type="math/tex"> y^{*} = e^{ax}[Q_l^{(1)}(x)\cos\beta x + Q_l^{(2)}\sin\beta x]x^k </script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{cases}
e^{ax}\text{照抄} \\[2ex]
l=\max\{m,n\} \\[2ex] 
Q_l^{(1)},Q_l^{(2)}\text{为x的两个不同的l次多项式} \\[2ex]
k = 
\begin{cases}
0,\ \alpha\pm\beta \text{i不是特征根} \\[2ex]
1,\ \alpha\pm\beta \text{i是特征根}
\end{cases}
\end{cases}
</script>
</p>
<h2 id="16">第16讲 无穷级数</h2>
<h4 id="1_10">1、正项级数</h4>
<p>
<script type="math/tex; mode=display">
\sum_{n=1}^\infty u_n,\ (u_n > 0)
</script>
</p>
<h5 id="1_11">1）比较判别法</h5>
<p>等比级数<br />
<script type="math/tex; mode=display">
\sum_{n=1}^\infty aq^{n-1}= \lim_{n\to\infty}\cfrac{a(1-q^n)}{1-q}
\begin{cases}
=\cfrac{a}{1-q}, & |q|\lt1 \\[2ex]
\text{发散}, & |q|\ge1
\end{cases}
</script>
</p>
<p>p级数<br />
<script type="math/tex; mode=display">
\sum_{n=1}^\infty \cfrac{1}{n^p}
\begin{cases}
\text{收敛}, & q\gt1 \\[2ex]
\text{发散}, & q\le1
\end{cases}
</script>
</p>
<p>广义p级数<br />
<script type="math/tex; mode=display">
\sum_{n=2}^\infty \cfrac{1}{n(\ln n)^p}
\begin{cases}
\text{收敛}, & q\gt1 \\[2ex]
\text{发散}, & q\le1
\end{cases}
</script>
<br />
交错p级数<br />
<script type="math/tex; mode=display">
\sum_{n=1}^\infty (-1)^{n-1}\cfrac{1}{n^p}
\begin{cases}
\text{绝对收敛}, & q\gt1 \\[2ex]
\text{条件收敛}, & 0\lt q\le1
\end{cases}
</script>
</p>
<p><strong>比较：设两个正项级数</strong><br />
<script type="math/tex; mode=display">
\sum_{n=1}^\infty u_n,\ (u_n > 0)，\sum_{n=1}^\infty v_n,\ (v_n > 0)
</script>
</p>
<p>有</p>
<p>
<script type="math/tex; mode=display">
\lim_{n\to\infty}\cfrac{u_n}{v_n}=l
\begin{cases}
0,                 & u_n 是 v_n 高阶无穷小，\sum_{n=1}^\infty v_n收敛，\sum_{n=1}^\infty u_n收敛\\[2ex]
0\lt c\lt +\infty, & u_n 是 v_n 同阶无穷小，\sum_{n=1}^\infty u_n，\sum_{n=1}^\infty v_n同敛散\\[2ex]
+\infty            & u_n 是 v_n 低阶无穷小，\sum_{n=1}^\infty v_n发散，\sum_{n=1}^\infty u_n发散
\end{cases}
</script>
</p>
<h5 id="2_9">2）比值判别法（达朗贝尔）</h5>
<p>
<script type="math/tex; mode=display">
\lim_{n\to\infty}\cfrac{u_{n+1}}{u_n}=\rho
\begin{cases}
\text{收敛}, & q\lt1 \\[2ex]
\text{发散}, & q\gt1 \\[2ex]
\text{失效}, & q=1
\end{cases}
</script>
</p>
<h5 id="3_8">3）根植判别式（柯西）</h5>
<p>
<script type="math/tex; mode=display">
\lim_{n\to\infty}\sqrt[n]{u_n}=\rho
\begin{cases}
\text{收敛}, & q\lt1 \\[2ex]
\text{发散}, & q\gt1 \\[2ex]
\text{失效}, & q=1
\end{cases}
</script>
</p>
<h4 id="2_10">2、交错级数</h4>
<p>
<script type="math/tex; mode=display">
\sum_{n=1}^\infty (-1)^{n-1}u_n,\ (u_n > 0)
</script>
</p>
<h5 id="1_12">1）莱布尼兹判别法</h5>
<p>
<script type="math/tex; mode=display">
\lim_{n\to\infty} u_n = 0\ 且\ u_n \ge u_{n+1}\Rightarrow 级数收敛
</script>
</p>
<h4 id="3_9">3、任意级数</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{n\to\infty}|u_n| 收敛                           & \Rightarrow \lim_{n\to\infty}u_n 绝对收敛 \\[1em]
\lim_{n\to\infty}|u_n| 发散，\lim_{n\to\infty}u_n 收敛 & \Rightarrow \lim_{n\to\infty}u_n 条件收敛
\end{split}\end{equation}
</script>
</p>
<ol>
<li>绝对收敛级数是收敛的，但收敛的级数不一定是绝对收敛级数。</li>
</ol>
<h4 id="4_6">4、收敛半径</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{n\to\infty}|\cfrac{a_{n+1}}{a_n}| &= \rho \\[2em]
\sum_{n=0}^{\infty}a_nx^n 的收敛半径R &= 
\begin{cases}
\cfrac{1}{\rho}, & \rho \ne 0, +\infty\\[2ex]
+\infty,         & \rho = 0 \\[2ex]
0,               & \rho = +\infty
\end{cases}
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>例题：【2020第17题】设数列 <script type="math/tex">\{a_n\}</script> 满足 <script type="math/tex">a_1=1,(n+1)a_{n+1}=(n+\cfrac{1}{2})a_n</script>，证明：当 <script type="math/tex">|x|<1</script> 时，幂级数 <script type="math/tex">\sum_{n=1}^\infty a_nx^n</script> 收敛，并求其和函数 <script type="math/tex">S(x)</script>
</p>
<p>1、证明，当 <script type="math/tex">|x|<1</script> 时，幂级数 <script type="math/tex">\sum_{n=1}^\infty a_nx^n</script> 收敛<br />
<script type="math/tex; mode=display">
\cfrac{a_{n+1}}{a_n}=\cfrac{n+\cfrac{1}{2}}{n+1} < 1
</script>
<br />
又 <script type="math/tex">a_1=1</script>，则数列 <script type="math/tex">\{a_n\}</script> 单调递减，且 <script type="math/tex">0<a_n<1 </script>，<script type="math/tex">|a_nx^n|<|x^n|</script>
</p>
<p>当 <script type="math/tex">|x|<1</script> 时，幂级数 <script type="math/tex">\sum_{n=1}^\infty x^n</script> 绝对收敛，故 <script type="math/tex">\sum_{n=1}^\infty a_nx^n</script> 绝对收敛（绝对收敛 <script type="math/tex">\to</script> 收敛）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
S(x)&=\sum_{n=1}^\infty a_nx^n \\[1em]
S'(x)&=(\sum_{n=1}^\infty a_nx^n)'=\sum_{n=1}^\infty na_nx^{n-1} \\
&=\sum_{n=0}^\infty (n+1)a_{n+1}x^{n} \\
&=a_1+\sum_{n=1}^\infty (n+1)a_{n+1}x^{n} \\
&=a_1+\sum_{n=1}^\infty (n+\cfrac{1}{2})a_{n}x^{n} \\
&=a_1+\sum_{n=1}^\infty na_{n}x^{n}+\cfrac{1}{2}\sum_{n=1}^\infty a_{n}x^{n} \\
&=a_1+x\sum_{n=1}^\infty na_{n}x^{n-1}+\cfrac{1}{2}\sum_{n=1}^\infty a_{n}x^{n} \\
&=a_1+xS'(x)+\cfrac{1}{2}S(x) \\[1em]
\end{split}\end{equation}
</script>
<br />
解微分方程：<script type="math/tex">(1-x)S'(x)-\cfrac{1}{2}S(x)=1</script>，即 <script type="math/tex">S'(x)-\cfrac{1}{2(1-x)}S(x)=\cfrac{1}{1-x}</script>
<br />
<script type="math/tex; mode=display">
p(x)=-\cfrac{1}{2(1-x)}， q(x)=\cfrac{1}{1-x}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
S(x)&=e^{-\int p(x)dx}(\int e^{\int p(x)dx}q(x)dx+C) \\
&=e^{-\int (-\frac{1}{2(1-x)})dx}(\int e^{\int (-\frac{1}{2(1-x)})dx}(\frac{1}{1-x})dx+C) \\
&=e^{-\frac{1}{2}\ln(1-x)}(\int e^{\frac{1}{2}\ln(1-x)}(\frac{1}{1-x})dx+C) \\
&=(1-x)^{-\frac{1}{2}}(\int(1-x)^{-\frac{1}{2}}dx+C) \\
&=(1-x)^{-\frac{1}{2}}(-2(1-x)^{\frac{1}{2}}+C) \\
&=-2+\cfrac{C}{\sqrt{1-x}}\\[1em]
由于S(0)&=0，得到C=2 \\[1em]
S(x)&=-2+\frac{2}{\sqrt{1-x}}
\end{split}\end{equation}
</script>
</p>
</blockquote>
<h2 id="17">第17讲 多元函数积分学的预备知识</h2>
<h4 id="1_13">1、方向导数</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{\partial u}{\partial \overrightarrow{l}}|_{P_0}
&= \lim_{t \to 0^+}\frac{u(x_0 + \Delta x, y_0 + \Delta y, z_0 + \Delta z) - u(x_0, y_0, z_0)}{\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}} \\[2ex]
&= \lim_{t \to 0^+}\frac{u'_x(P_0)\Delta x + u'_y(P_0)\Delta y + u'_z(P_0)\Delta z - o(t)}{\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}}\\[3ex]
&= u'_x(P_0)\cos\alpha + u'_y(P_0)\cos\beta + u'_z(P_0)\cos\gamma
\end{split}\end{equation}
</script>
</p>
<h4 id="2_11">2、梯度</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\mathbf{grad}}\ u|_{P_0} = (u'_x(P_0), u'_y(P_0), u'_z(P_0))
</script>
</p>
<p>梯度与方向导数的关系<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{\partial u}{\partial \overrightarrow{l}}|_{P_0} &= u'_x(P_0)\cos\alpha + u'_y(P_0)\cos\beta + u'_z(P_0)\cos\gamma \\
&= \overrightarrow{\mathbf{grad}}\ u|_{P_0} * (\cos\alpha,\cos\beta,\cos\gamma)
\end{split}\end{equation}
</script>
<br />
<strong>梯度的方向</strong>与<strong>取得最大方向导数的方向</strong>一致，最大值为<strong>梯度的模</strong></p>
<h4 id="3_10">3、散度</h4>
<p>向量场 <script type="math/tex"> \overrightarrow{A}(x,y,z) = P(x,y,z)\overrightarrow{i}+Q(x,y,z)\overrightarrow{j}+R(x,y,z)\overrightarrow{k} </script> ，则<br />
<script type="math/tex; mode=display">
div\ \overrightarrow{A} = \cfrac{\partial P}{\partial x} + \cfrac{\partial Q}{\partial y} + \cfrac{\partial R}{\partial z}
</script>
</p>
<ol>
<li>
<script type="math/tex"> div\ \overrightarrow{A} </script>  表示场在 <script type="math/tex"> (x,y,z) </script> 处源头的强弱程度。</li>
<li>若  <script type="math/tex"> div\ \overrightarrow{A} = 0 </script>  在场内处处成立，则称A为<strong>无源场</strong>。</li>
</ol>
<h4 id="4_7">4、旋度</h4>
<p>向量场 <script type="math/tex"> \overrightarrow{A}(x,y,z) = P(x,y,z)\overrightarrow{i}+Q(x,y,z)\overrightarrow{j}+R(x,y,z)\overrightarrow{k} </script> ，则<br />
<script type="math/tex; mode=display">
\overrightarrow{\mathbf{rot}}\ \overrightarrow{A} = 
\begin{vmatrix}
\overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k} \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix}
</script>
</p>
<ol>
<li>
<script type="math/tex"> \overrightarrow{\mathbf{rot}}\ \overrightarrow{A} </script>  表示场在 <script type="math/tex"> (x,y,z) </script> 处最大旋转趋势的度量。</li>
<li>若  <script type="math/tex"> \overrightarrow{\mathbf{rot}}\ \overrightarrow{A} = 0 </script>  在场内处处成立，则称A为<strong>无旋场</strong>。</li>
</ol>
<h3 id="_9">空间几何</h3>
<h4 id="1_14">1、空间曲线的切线与法平面</h4>
<h5 id="1_15">1）参数方程给出曲线</h5>
<p>曲线在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的切向量<br />
<script type="math/tex; mode=display">
\overrightarrow l = (x'(t_0), y'(t_0), z'(t_0))
</script>
<br />
切线方程<br />
<script type="math/tex; mode=display">
\cfrac{x-x_0}{x'(t_0)} = \cfrac{y-y_0}{y'(t_0)} = \cfrac{z-z_0}{z'(t_0)}
</script>
<br />
法平面方程<br />
<script type="math/tex; mode=display">
x'(t_0)(x-x_0) + y'(t_0)(y-y_0) + z'(t_0)(z-z_0) = 0
</script>
</p>
<h5 id="2_12">2）用方程给出曲线</h5>
<p>
<script type="math/tex; mode=display">
\overrightarrow l = (1, y'(x_0), z'(x_0))
</script>
</p>
<p>&hellip;</p>
<h4 id="2_13">2、空间曲面的切平面与法线</h4>
<h5 id="1_16">1）隐式给出曲面</h5>
<p>曲面在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的法向量<br />
<script type="math/tex; mode=display">
\overrightarrow n = (F'_x|_{P_0}, F'_y|_{P_0}, F'_z|_{P_0})
</script>
<br />
法线方程<br />
<script type="math/tex; mode=display">
\cfrac{x-x_0}{F'_x|_{P_0}} = \cfrac{y-y_0}{F'_y|_{P_0}} = \cfrac{z-z_0}{F'_z|_{P_0}}
</script>
<br />
法平面方程<br />
<script type="math/tex; mode=display">
F'_x|_{P_0}(x-x_0) + F'_y|_{P_0}(y-y_0) + F'_z|_{P_0}(z-z_0) = 0
</script>
</p>
<h5 id="2_14">2）显式给出曲面方程</h5>
<p>曲面在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的法向量<br />
<script type="math/tex; mode=display">
\overrightarrow n = (z'_x(x_0,y_0), z'_y(x_0,y_0), -1)
</script>
<br />
&hellip;</p>
<h5 id="3_11">3）用参数方程给出曲面</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
x = x(u,v) \\[2ex]
y = y(u,v) \\[2ex]
z = z(u,v) 
\end{cases}
</script>
</p>
<p>
<script type="math/tex"> u </script> 曲线在 <script type="math/tex"> P_0 </script> 处的切向量为 <script type="math/tex"> \overrightarrow{l_1} = (x'_u, y'_u, z'_u)|_{P_0} </script>
</p>
<p>
<script type="math/tex"> v </script> 曲线在 <script type="math/tex"> P_0 </script> 处的切向量为 <script type="math/tex"> \overrightarrow{l_2} = (x'_v, y'_v, z'_v)|_{P_0} </script>
</p>
<h2 id="18">第18讲 多元函数积分学</h2>
<h4 id="1_17">1、二重积分</h4>
<p>几何意义：曲顶柱体的体积<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iint_D f(x,y)d\sigma &= \int_{a}^{b} dx \int_{y_1(x)}^{y_2(x)} f(x,y)dy \\[1ex]
&= \int_{a}^{b} dy \int_{x_1(x)}^{x_2(x)} f(x,y)dx
\end{split}\end{equation}
</script>
</p>
<h4 id="2_15">2、三重积分</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iiint_\Omega f(x,y,z)dv &= \int_{a}^{b} dx \int_{y_1(x)}^{y_2(x)} dy \int_{z_1(x,y)}^{z_2(x,y)} f(x,y,z)dz \\[1em]
&= \int_{a}^{b} dz \iint_{D_{xy}} f(x,y,z) dxdy \\[3em]
& \begin{cases}
x = r\cos\theta \\[2ex]
y = r\sin\theta \\[2ex]
z = z \\[2ex]
dxdydz = r\ \ dr d\theta dz
\end{cases} \\[1em]
(柱面坐标系) &= \iiint_\Omega f(r\cos\theta,\ r\sin\theta,\ z)r\ \ dr d\theta dz \\[3em]
& \begin{cases}
x = r\sin\phi\cos\theta \\[2ex]
y = r\sin\phi\sin\theta \\[2ex]
z = r\cos\phi \\[2ex]
dxdydz = r^2\sin\phi\ \ d\theta d\phi dr
\end{cases} \\[1em]
(球面坐标系) &= \iiint_\Omega f(r\sin\phi\cos\theta,\ r\sin\phi\sin\theta,\ r\cos\phi)r^2\sin\phi\ \ d\theta d\phi dr
\end{split}\end{equation}
</script>
</p>
<h4 id="3_12">3、第一型曲线积分（数量值函数在曲线上的积分）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_Lf(x,y)ds &= \int_\alpha^\beta f[\phi(t),\psi(t)]\sqrt{\phi'^2(t)+\psi'^2(t)}dt \\ 
&= \int_\alpha^\beta f[x,y(x)]\sqrt{1+y'^2(x)}dx
\end{split}\end{equation}
</script>
</p>
<h4 id="4_8">4、第二型曲线积分（向量值函数在曲线上的积分）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_L\overrightarrow f(x,y)\overrightarrow{dr} &= \int_L Pdx + Qdy \\[1ex]
&= \int_{t_!}^{t_2} \{\ P[\phi(t),\psi(t)]*\phi'(t) + Q[\phi(t),\psi(t)]*\psi'(t)\ \}dt
\end{split}\end{equation}
</script>
</p>
<p>第一型曲线积分与第二型曲线积分的关系<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\int_L Pdx + Qdy &= \int_{t_!}^{t_2} (P\cos\alpha + Q\cos\beta)ds \\[3em]
方向余弦\ \cos\alpha &= \cfrac{\phi'(t)}{\sqrt{\phi'^2(t)+\psi'^2(t)}} \\[1ex]
\cos\beta &= \cfrac{\psi'(t)}{\sqrt{\phi'^2(t)+\psi'^2(t)}}
\end{split}\end{equation}
</script>
</p>
<h4 id="5_5">5、格林公式（第一型曲线积分与二重积分）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\oint_LPdx+Qdy &= \iint_D(\cfrac{\partial Q}{\partial x} - \cfrac{\partial P}{\partial y})dxdy
\end{split}\end{equation}
</script>
</p>
<h4 id="6_4">6、斯托克斯公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\oint_LPdx+Qdy+Rdz
&= \iint_D \begin{vmatrix}
dydz & dzdx & dxdy \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix} &\text{(第二型曲面积分)} \\[1ex]
&= \iint_D \begin{vmatrix}
\cos\alpha & \cos\beta & \cos\gamma  \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix} dS \ \ \ \ &\text{(第一型曲面积分)}
\end{split}\end{equation}
</script>
</p>
<h4 id="7">7、第一型曲面积分（数量值函数在曲面上的积分）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iint_\Sigma f(x,y,z)dS &= \iint_{D_xy} f(x,y,z(x,y))\sqrt{1+z_x'(x,y)+z_y'(x,y)}dxdy
\end{split}\end{equation}
</script>
</p>
<h4 id="8">8、第二型曲面积分（向量值函数在曲面上的积分）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iint_\Sigma \overrightarrow F(x,y,z)\overrightarrow{dS} =& \iint_\Sigma Pdydz + Qdzdx + Rdxdy \\[1ex]
=&\ \ \ \  \iint_{D_{yz}} P(x(y,z),y,z)dydz \\[1ex] &+ \iint_{D_{zx}} Q(x,y(z,x),z)dzdx \\[1ex] &+ \iint_{D_{xy}} R(x,y,z(x,y))dxdy
\end{split}\end{equation}
</script>
</p>
<p>第一型曲面积分与第二型曲面积分的关系<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&(第一型曲面积分) \\[1ex]
\iint_\Sigma \overrightarrow F(x,y,z)\overrightarrow{dS}&= \iint_\Sigma (P\cos\alpha + Q\cos\beta + R\cos\gamma)dS \\[1ex]
&= \iint_\Sigma (P\cfrac{\cos\alpha}{\cos\gamma} + Q\cfrac{\cos\beta}{\cos\gamma} + R)\cos\gamma \ dS \\[1ex]
&= \iint_\Sigma (P(-z_x') + Q(-z_y') + R)dS \\[3em]
方向余弦\ \cos\alpha &= \cfrac{-z_x'}{\sqrt{1+z_x'^2+z_y'^2}} \\[1ex]
\cos\beta &= \cfrac{-z_y'}{\sqrt{1+z_x'^2+z_y'^2}} \\[1ex]
\cos\gamma &= \cfrac{1}{\sqrt{1+z_x'^2+z_y'^2}}\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex"> \Sigma </script> 前右上侧为 <script type="math/tex"> + </script> ，后左下侧为 <script type="math/tex"> - </script>
</p>
</blockquote>
<h4 id="9_1">9、高斯公式（第一型曲面积分与三重积分）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\iiint_\Omega(\cfrac{\partial P}{\partial x} + \cfrac{\partial Q}{\partial y} + \cfrac{\partial R}{\partial z})dv &= \oiint_\Sigma Pdydz+Qdzdx+Rdxdy \\[1ex]
&= \oiint_\Sigma (P\cos\alpha + Q\cos\beta + R\cos\gamma)dS
\end{split}\end{equation}
</script>
</p>
<h5 id="_10">实际应用</h5>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学考研-高等数学.assets/IMG_0092.jpg" alt="IMG_0092" style="zoom:20%;" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(a)\iint_{D}f(x,y) &= \int_\alpha^\beta d\theta\int_{r_1(\theta)}^{r_2(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D外部）}\\
(b)\iint_{D}f(x,y) &= \int_\alpha^\beta d\theta\int_0^{r(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D边界上）} \\
(c)\iint_{D}f(x,y) &= \int_0^{2\pi}d\theta\int_0^{r(\theta)}f(r\cos\theta,r\sin\theta)rdr\text{（极点O在区域D内部）}
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p><strong>在求解时应该充分利用函数的对称性，如奇对称、偶对称、<u>轮换对称性</u>&hellip;</strong></p>
</blockquote>
</div>
<div id="nav">
  <div class="navigation">
  <ul class="pure-menu-list">
    <li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.25em;" href="#_1">第一部分 高等数学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#1">第1讲 高数预备知识</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1-a_1-dd-neq-0">1. 等差数列（首项  <script type="math/tex"> a_1 </script> ，公差  <script type="math/tex"> d(d \neq 0) </script>  ）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2-a_1-rr-neq-0">2. 等比数列（首项  <script type="math/tex"> a_1 </script> ，公比  <script type="math/tex"> r(r \neq 0) </script> ）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3">3. 三角函数基本关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_1">1）倍角公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2">2）和差公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_1">3）积化和差公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#4">4）和差化积公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_1">4. 因式子分解公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#5">5. 常用不等式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#2_1">第2讲 函数极限</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#0">0. 基本极限公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_2">1. 洛必达法则</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_2">2. 泰勒公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#5_1">定理5（泰勒公式）（忘记常用泰勒级数展开式，可以用下面的公式求）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#_2">常用泰勒级数展开式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_2">3. 无穷比介小</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_2">4.  无穷小的运算规则</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#5_2">5. 常用的等价无穷小</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#6">6. 函数的连续与间断</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#_3">补充：渐近线</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_3">1、水平渐近线和铅直渐近线</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2x">2、斜渐近线的正确求法(在x趋向于无穷时)</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#4_3">第4讲 一元函数微分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_4">1. 导数定义</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_3">2. 导数与微分的计算</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_3">3. 可微、可导、连续、可积的关系</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#4_4">4. 莱布尼茨公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#5_3">5. 泰勒公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#6_1">6. 参数方程</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#6_2">第6讲 一元函数微分学的应用</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_5">定理1（费马定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_4">定理2（罗尔定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_4">定理3（拉格朗日中值定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#4_5">定理4（柯西中值定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#5_4">定理5（泰勒公式）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#6_3">定理6（积分中值定理）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#9">第9讲 一元函数积分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_4">“点火公式”</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_5">伽马函数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_6">分部积分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_7">基本公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#13">第13讲 多元函数微分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1_6">1. 导数与微分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_7">1）偏导数的定义公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_5">2）二元函数微分的定义</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_5">3）复合函数求导法</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_6">2. 二元函数极值</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_6">3. 条件极值</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#fxy">（二元）函数 <script type="math/tex">f(x,y)</script> 在条件</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#fxyz">（三元）函数 <script type="math/tex">f(x,y,z)</script> 在条件</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#15">第15讲 微分方程</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_8">1. 一阶微分方程的求解</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_9">1）可分离变量型</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_7">2）齐次型</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_7">3）一阶线性型</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_8">伯努利方程</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_8">2. 二阶微分方程的求解</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#16">第16讲 无穷级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_10">1、正项级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_11">1）比较判别法</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_9">2）比值判别法（达朗贝尔）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_8">3）根植判别式（柯西）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_10">2、交错级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_12">1）莱布尼兹判别法</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_9">3、任意级数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_6">4、收敛半径</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#17">第17讲 多元函数积分学的预备知识</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_13">1、方向导数</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_11">2、梯度</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_10">3、散度</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_7">4、旋度</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#_9">空间几何</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_14">1、空间曲线的切线与法平面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_15">1）参数方程给出曲线</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_12">2）用方程给出曲线</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_13">2、空间曲面的切平面与法线</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#1_16">1）隐式给出曲面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#2_14">2）显式给出曲面方程</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#3_11">3）用参数方程给出曲面</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#18">第18讲 多元函数积分学</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_17">1、二重积分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_15">2、三重积分</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_12">3、第一型曲线积分（数量值函数在曲线上的积分）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4_8">4、第二型曲线积分（向量值函数在曲线上的积分）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#5_5">5、格林公式（第一型曲线积分与二重积分）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#6_4">6、斯托克斯公式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#7">7、第一型曲面积分（数量值函数在曲面上的积分）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#8">8、第二型曲面积分（向量值函数在曲面上的积分）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#9_1">9、高斯公式（第一型曲面积分与三重积分）</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 2.25em;" href="#_10">实际应用</a>
</li>

  </ul>
</div>

</div>
 
    </div>
  </div>
  <div id="footer">
    <div class="legal pure-g">
  <div class="pure-u-1 u-sm-1-2">
    <p class="legal-license"><a href="https://beian.miit.gov.cn/#/Integrated/index">浙ICP备2020038748号</a></p>
  </div>
  <div class="pure-u-1 u-sm-1-2">
    <p class="legal-links"><a href="https://github.com/zromyk">GitHub</a></p>
    <p class="legal-copyright">Copyright © 2021 Wei Zhou. 保留所有权利。</p>
  </div>
</div>
  </div>
  <script src='/style/latest.js?config=TeX-MML-AM_CHTML'></script>
  <script src="https://cdn.bootcss.com/jquery/3.2.1/jquery.min.js"></script>
  <script src='/style/Valine.min.js'></script>
  <script src="https://apps.bdimg.com/libs/highlight.js/9.1.0/highlight.min.js"></script>
  <script type="text/javascript">
    hljs.initHighlightingOnLoad();
  </script>
  <script src="https://cdn.geogebra.org/apps/deployggb.js"></script>
  <script src="https://cdn1.lncld.net/static/js/2.5.0/av-min.js"></script>
  <script src='/style/readTimes.js'></script>
</body>
</html>
